The set $S=\{1,2,3,\dots ,12\}$ is to be partitioned into three sets $A,B,C$ of equal size. Thus, $A\cup B\cup C=S,A\cap B=B\cap C=A\cap C=\varphi$. The number of ways to partition $S$ is:

The set $S$ has 12 elements. We need to partition it into three sets $A,B,C$ of equal size, meaning each set must have $12/3=4$ elements.Step 1: Choose 4 elements for set $A$ from 12 elements: ${}^{12}{C}_4$ ways.Step 2: Choose 4 elements for set $B$ from the remaining 8 elements: ${}^8{C}_4$ ways.Step 3: Choose 4 elements for set $C$ from the remaining 4 elements: ${}^4{C}_4$ ways.Total number of ways = ${}^{12}{C}_4\times {}^8{C}_4\times {}^4{C}_4$ $
$=\frac{12!}{4!8!}\times \frac{8!}{4!4!}\times \frac{4!}{4!0!}$ $
$$=\frac{12!}{4!4!4!}=\frac{12!}{(4!{)}^3}$$
Note: Since the sets $A,B,$ and $C$ are distinct (labeled), we do not divide by $3!$.